CON IGYT - Conicyt

'
CON IGYT
COMISIÓN NACIONAL DE INVESTIGACIÓN
CIENIIFICA Y IICNOLÓGICA
GOBIERNO DE CHILE
ni
COMIS1ON NACIONAL DE INVESTIGACION CIENCIA Y TECNOLOGIA
VERSION OFICIAL
FECHA: 30/11/2009
PROYECTO INICIACION N°11060538
INVESTIGADOR RESPONSABLE: BARTLOMIEJ MARJUSZ SKORULSKJ
FONDO NACIONAL DE DESARROLLO CIENTIFICO Y TECNOLOGICO (FONDECYT) Bernarda Morín 551, Providencia - casilla 297- y, Santiago 21
Telefono: 435 43 50 FAX 365 4435
Email: infoxmes.fondecytconicyt.c1
INFORME FINAL
PROYECTO FONDECYT INICIACION
AÑO ETAPA: 2008
DURACIÓN: 3 años
N° PROYECTO: 11060538
TíTULO PROYECTO: INVARIANT MEASURES AND THERMODYNAMIC FORMALISM FOR MEROMORPHIC
FUNCTIONS
DISCIPLINA PRINCIPAL: SISTEMAS DINAMICOS
GRUPO DE ESTUDIO: MATEMATICAS
INVESTIGADOR(A) RESPONSABLE: BARTLOMIEJ MARI USZ SKORULSKI
DIRECCIÓN : Altos del Mar '1001 dep. 151 torre B
COMUNA:
CIUDAD: ANTOFAGASTA
REGIÓN: II REGION
FONO: 56-55-355526
EMAIL: bskorulski@ucn.Cl
INFORME
OBJETIVOS
Cumplimiento de los Objetivos planteados en el Proyecto. Recuerde que los objetivos del proyecto no se refieren a listar actividades
desarro!IaUaS sino a ios oojettvus uaIIu.
OBJETIVOS
Undcrstanding the role of conformal measures in
the dynamics of transcendental functions and the
conditions which implies their existence
Developing thc theory of the real transformation
2
CUMPLIMIENTO
PARCIAL
written a paper containing resuits which 1 was
NO
with fiat critical point
Moving forward tbe study of the exponential
3
FUNDAMENTO
2 papers were finished and published, however
they contain only partial results. Also J. Kotus has
PARCIAL
working on.
With C. Vasquez we have written a preprint,
hower N. Dobbs has got more general result
before us
With N. Dobbs we solved an open question
related to the existence of acip for Misiurewicz
maps. Now we are working on slowly escaping
maps, which are already very compilcated.
family
Therefore CE-like conditiori is still out of out
4
Developing random dynamics for transcendental
PARCIAL
maps
range.
Before developing non-rational mcromorphic
dynamics we started lo work on random
expanding maps en compact fibers. With V.
Mayer and M. Urbanski we have written a long
paper on that (submitted). In this paper we gaye
an answer to two conjectures. 1 was also working
with G. lommi en similar problem in more
specific situation. Now with Mayer and lirbanski
5
Proving stochastic stability for maps from 2
NO
we work on meromorphic random dynamics.
Since the point 2 was not done this as well.
Otro(s) aspecto(s) que lid, considere importante(s) en la evaluación del cumplimiento de objetivos planteados en la propuesta
original o en las modificaciones autorizadas por los Consejos.
¡ think the niost interesting part of this project was part related to point 4. When with V. Mayer and M. Urbanski we started to work
on Random Non-Rational Maps we realized that There are still Iots of problems in the case of compact fibers (in transcendental
settings fibers are equal to the complex plane and transformations have essential singularity at inlinity). We have found out that
there are still sorne important question left without answer in the case of rational dynamics. Wc have given an answer to these
conjectures. Moreover, we have found out that there is mostly not work done in the subject of multifractal spectrum of randoni
maps. We also have spent lots of time trying to get a sufficiently good results in this subject. With G, lommi we also worked on the
same problem in a very special case of random expansion of real nurnbers. We were trying (o find a direct formula for the level seis,
which is a special case of radnom expanding maps. We however have realized that we can formulate and solve this problem in a
completely different language of Cantor series. Now with V. Mayer an M. Urbanski we are working on transcendental case. Work is
still in progress but promising.
With the respect to point 3 we (with N. Dobbs) we obtained a very important result about not existence of acip. On one hand, this
was an answer to an open question, on the other these result has opened a completely new way of working with meromorphic
dynamics which seems to simplified a lot many partial results 1 havc obtained. With N. Dobbs and M. Urbanski we are now working
on these problems.
With respect to point 1 1 have got a partial results and working on more general settings while J. Kotus have published a paper
which contained the missed case. Very similar thing has oecurred with work related the point 2. This time N. Dobbs have informed
me about generalization of results 1 had. 1 also have not succeed in work on other issues addrcssed in this point and poinl 5 as well.
RESULTS OBTAINED:
Fui cadi specific gua! 7 describe or surninarize the results obtained. Relate cadi une tu
work already published and/or inanuscripts suhrnitted. In the Annex sectiori include
additional inforination deemed pertirient aiid relevant to the evaluation process. The
maximum length for this section is 5 pages. (Aria! or Verdana, font size
10).
Thermodynamic formalism for meromorphic functions
In [6] with lon Coiculescu we construct a conforunal uneasure for expanding eiitire
functions in farnily of function f of the forrn f(z) = R(exp(z)), where R is a ratiouial
funetion and 1. does no llave a finite asyunptutic value. Then Wc show the existence of
invariant uneasure which is equivalent to the conforinal ouie. This WS a part ial answer
to the questioui of existeuice of invariant measures absolutely continuous with the
respect to conformal measure for hyperbolic inaps of the forin R(exp(z)). Together
with non-euitire case lorni Kotus' article [13] this solved coinpletely the probleun.
In [6] we also get a version of Bowen formula. This formula plays important role
in our second paper [5]. In this paper we proved real-analycity of the Hausdorff
diunension of the radial Julia set. This paper gives an important foundation for
study sume non-hyperbolic non-entire fuuictions. With Janina Kotus we studied a
one-parainet.er fainily which unembers has ami a.syinptotic value eventually mapped
onto infinity. For this family we proved that the Hausdorff dimension of the radial
Julia set is also real-analytic. Tui order to get it we have to also showed a version
uf Thcuremn of Mae-Sad-Sullivami for rational inaps or Ercmenko-Lyubich fui entire
transformations. However, in this preprint we went ori a very couuiplicated ruad in
order to overcome a difficulty related to the periodicity of considered function. Now,
a new technique related of fice set, would make the prove rnuch easier. And we are
working un that (see the next section).
Exponential family
With Neil Dobbs we were studing the exisitence of invariant ineasures absoultely
[7] we solved an open (for a
continuuus with respect to the Lebesgue measue. In [71
suprisingly long time) question. Wc shuwed tlmat Misiurewicz Expunential Maps
cannot have aii finite absolutely invariant measure. Wc used a interesting techinique
of uuice sets developed by Juan Rivera. This was the first time that somnebody applied
this for transcendental maps. It turned out that, J. Kotus and G. Swiatek proved
the sanie result. Huwever their pruof is defiuuite!y longer and more counplicatcd.
The existence of ahsolutely invariant measures is know for Misiurewicz type transformnations and uiot existence is known for exponential maps with the orbit of zero
of the first return map to the left half plane escaping very fast torwards infinity. So
with N. Dobbs we are wurkiuig un puints which has a differcnt escaping rate. Let
1
y,-, := IT n(0)j where T is the return rnap to the left lialf of the plane. We proved
that, if v > O(logn), then absoultely coritinuous irivariant measures caniiot cxist,
and if y,., < O(loglogn) and v,. > d > 1 for ah largo n, then a absolutely continuous
invariant measurc exist, however is not finite. Tu the prove of the theoreni we used
techuiiques similar to studying the Browriian motion. It is however still rexnairis open
if ono can find a coiiditioii which can guarantee that une of the both cases exist. Unfortunateiy the calculation are a bit complicated and we are still working un puttiiig
everything together.
Random Dynamical Systems
In [14] with Volker Mayor aud Mariusz Urbaiiski we develop the thermodynamical
formahisni for measurable expandmg raridom mappings. This theory then is appiied
iii the context of conforinal expanding randoin mappings where we deal with the
fractal gcometry.
Measurable expanding random mappings. First we define measurable expanding random maps. The raiidomness is inodeled by an invertible ergodic trarisforniation O of
a probability space (X, 8, m). Wc investigate the dynamics of compositions
7=TCr-l (x)
O
... OTx ,n>1,
where the T : jx - 3(x) (x E X) is a distance expanding Inapping. These inaps
are only siipposed to be nieasurably cxpanding iii tite sense titat thcir expanding
constant is ineasurable and a.e. Yx > 1 or f log 'Yx dm(x) > 0.
Iii so general setting we first build the therinodynainical formahisrn for arbitrary
Holder continuous potentials W , . We show, in particular, the existence, uniqueness
and ergochicity of a faiiiily of Cibbs measures {'-'x}x€X. Followiiig ideas of Kifer [10],
these measures are flrst produced in a pointwise nianner and then we carefuhly Check
their measurabihity. Iii Reinark 3.3 and 3.4 of [9] Kifer aud Khaiiiri indicated a
possibihty tu build thermodyiiarnic forinahisin for raridorn distance expandiiig Iriaps.
They saw tite obstados to huild sucli formalisni ni tite lack of Markov partit.ioris
sirice they needed tliein iii order to use syrnbohic representation froin [1]. Iii our
approach we do not need any Markov partitions or (even auxihiary) syinboh dynainics.
Moreover, itt [1] and [9] all fibers are iequired to be contained iii tite sanie compact
mctric sparc. Wc do not even need these fibers to be coritained in une metric sparc.
Our resuhts contain those in [1] and in [10] (see also the expository artiche [121).
Throughout the entire paper where it is possibhe we avoid, iii hiypotheses, absohute
coiistants. Our feelirig is that iii the eontext of random systems ah (or at least
as mnany as possihle) al)soiute constants appearing in determiiiistic systems should
becoine measurable functions. With this respect time thermodynamnical foriiialism
devehoped in here represents also, up to our knowledge, new achievemnents in the
2
theory of random symbol dynainics or smooth expanding random inaps acting 011
Rieinaririiaii inaiiifolds.
Wc show that this convergerice is exponential, which implies exponexitial decay of
correlations. These results precede investigations of a pressure funetion x P(Ø)
which satisfies the property
1Jo()(T(A))
=
JA
edu
where A is any measurable set such that TXIA is injective. The integral, against
t.he ineasure rn on the base X, of this function is a central parameter eP(p) of
randoin systems called the expected pressure. If the poteiitial 0 depends analytically
oil parameters, we show that the expected pressure also hehaves real analytically.
Wc would like tu rnentiun that, contrary to the deterrriinistic case, the spectrai gap
inethods (10 not work iii the randoni setting. Our proof utilizes the conccpt of complcx
cones introduced by Rugh in [15].
Conformal random systems Wc then appiy the aboye results mainly to investigate
fractal pruperties of fibers of conformal random systems. Tliey inelude Hausdorff
dimension, Hausdorff and packing measures, as well as multifractal analysis. First,
we establisli a version of Bowens formula (obtairied in somewhat different context
iii [21) showing that the Hausdorff diinension of alinost every fiber ¿T1 is equal to h,
the only zero of the expected pressure SP(), where O t = — tiog f'j and 1 E R. Tuco
we analyze the behavior of h dimensional Hausdorff and packi ig nieasurcs, It turneci
out that thc random d iiamical systems split into two categories. Systems from the
first category, rather exceptional, behave like deterrninistic systems. Wc cali thern,
therefore, quasi-deterrninistic. For thern tlie Hausdorff and packing measures are
finite and positive. Other systems, called essentially random, are rather generic. For
them the h-dimensional Hausdorff measure vanishes whuie the h-packing measure is
infinite. This, iii particular, refutes the conjecture stated by Bogenschütz and Ochs
iii [21 tliat the h—dimensional Flausdorff mneasure of fibers is always positive and limite.
In fact, tlic (iist.inction hetween the quasi-determinist.ic and the essentially ranclom
systems is deterrnined by the behavior of the Birkhoff sums
P)
= P0
i(ç) + ... + Pl)
of the pressure funetion for potential O h = —h log If'L If tliese sums stay bounded
theu we are in the quasi-determninistie case. Qn the other harid, if these sumos are mcifuer bounded below nor ahoye, the systemn is calied essentialiy random. rrIlc behavior
of P, being random variables defined on X. the base map for our skew product map,
is often governed by stochastic theorems such as tlie law of the iterated logarithrn
3
whenever it holds. This is the case for our priinary exainples, namely conforinal DGsystems aud elassical conformal randoin systems. Wc are then iii position to state
that the quasi-deterrninistic systems correspond to rather exceptional case where the
asymptotic variance a2 = O. Otlierwise the system is essent.ial.
The fact that Hausdorff measures iii the Hausdorff dimensiori vaiiisli has further
st.riking geometric consequenees. Namely, almost all fibers of an essential conforixial
raridoin system are not hi-Lipschitz equivalent to any fiber of any quasi-cleterministie
or determiuistic conforinal expanding system. In consequence almost every fiber of
an essentially randorn systern is iiot a geometric circie nor even a piecewise analytie
curve. Wc then show that thesc results do hoid for many explicit raxidwn dyiiamical
systems, such as coiiformal DG-systems, classical conforinal randoin systems, and,
perhaps rnost importantly, Brück and Bürger polynornial systems. As a coxisequence
of the techniques we llave developed, we positively answer the question of Brück and
Bürger (see [4] and Question 5.4 iii [3]) of whether the Hausdorff dimension of ahnost
alT naturaily define(¡ random Julia set is strictly larger than 1. We also show that
iii this sanie settixig thc Hausdorff dimeiision of alinost all Julia sets is strictly less
than 2.
If the systein is iii addition uniforinly expanding then we provide real analytieity of
t.he pressure fuxiction. As a consequence and via Legendre transformation we ohtain
real analyticity of the inultifractal spectruxn.
Multifractal spectrum. Concerning the multifractal spectrum of Gibbs ineasures on
fibers, we show that the multifractal forinalisin is valid, j.c. the inultifractal spectruin
is Legendre conjugated to a ternperature function. As usual, t.he t.ernperature functioxi is iinplicitly given ni terms of the expected pressure. Here, the inost important,
although perhaps not most strikingly visible, issue is to make sure thiat there exists
a set Xma of fuhi measure iii the base such that the niultifractal foi'malism works for
all x E X,,,, Hausdorff Diinension of Cantor Series. A very special exarnple of a Randoin Dynamical Systein is related te Cantor Series. Tuis a work we did with Godofredo
]S
loinnu in 181.
Let A = {b}n E N be a sequence of positive integers such that b n e N \ {1}. Every
real number x e [0, 11 can be written as
r;
where e 72 (x) E {O, 1, . . . ,
- 1}. Wc write
X
= [ e 1 (x)e 2 (x) . . . €(x) . . .
4
and cail it the A— Cantor Series of x with respect to the base A = {b, L } flc. For
every base A the Cantor series is unique exeept for a countable number of points.
Note that if A is tlie sequence such that for every n E N WC liave b = b, theri tlie
A—Cantor series corresponds to the base b expansion of x E [0, 1].
In [11] Y. Kifer obtained a variational formula for the Hausdorff dimension of tlie
set of poin1s for which the frcquenies of tlie digits in the Cantor series expansion is
givell. Wc present a slightly different approach to this problein that allow jis to solve
the variational problein of Kifer's formula. His formula involves a supremuin while
our can be caleulated directly.
References
[1] Thomas Bogerischütz and Volker Mathias Gundlach. Ruelle's transfer operator
for randoiri subshifts of finite type. Ergodic Thcory Dyrta'rn. Systems, 15(3):413447, 1995.
[21 Thomas Bogeiiscliiitz and Gunter Ochs. The Hausdorff dimension of conformal
repeliers under randoiii perturbation. Nonlznearzty, 12(5):1323-1338, 1999.
[3] Rainer Brück. Geomnetric properties of Julia sets of the comnposition of poiynomials of the form z2 + c. Pacific J. Math., 198(2):347-372, 2001.
[4] Rainer Brück and Matthias Büger. Generalized iteration. Comput. Methods
Funct. Theory, 3(1-2) :201-252, 2003.
[5] Ion Coiculescu and Bartlorniej Skorulski. Perturbations in t.hc Speiser class.
Rocky Mountain J. Math., 37(3):763-800. 2007.
[6] Ion Coiculescu and Bartloiniej Skorulski. Therinodyiiamxiic foimnalismn of transcendental entire maps of finite singular type. Monatsh. Math., 152(2):105-123,
2007.
[7] Neil Dobbs amid Bartlomiej Skorulski. Non-existence of absolutely continuous
invariant probabilities for exponential maps. Fund. Math., 198(3):283-287, 2008.
[8] Godofredo lomnmi and Bartlomniej Skorulski. Hausdorff dimnension of cantor series. submitcd.
[9] K. Khanin and Y. Kifer. Thermodymiamic formnalism for randoin transforinatiomis
and statistical meclianies. In Sina"s Moscow Semina, oit Dy'namical Systems,
5
volurne 171 of Amer. Math. Soc. Transi. Ser. 2, pages 107-140. Amer. Math.
Soc., Providencu, Rl, 1996.
[101 Yuri Nifer. Equilibriuin states for raiidoin expanding transforrnations. Random
Comput. Dynam,, 1(1):1-31, 1992/93.
[11] Yuri Nifer. Fractal dirnensions and random transforinations. Trans. Amer.
Math. Soc., 348(5): 2003-2038 1996.
[12] Yuri Nifer and Pei-Dong Liu. Random dynamics. Iii Handbook of dynamical
systems. Vol. iB, pages 379-499. Elsevier B. V., Amsterdam, 2006.
[13] Janimia Kotus. Proljabilistic invariant mncasures for non-entire funct.iuns with
asymptotic values imiapped onto oc. Illinois J. Math., 49(4):1203-1220 (eleetronic), 2005.
[14] Volker Mayer, Urhaúski Mariusz, aud Bartlomniej Skorulski. Distance expanding random inappings, therinodynamic formalismn, gibhs measures, and fractal
geometry. submnit.ed.
151 Hans Hcnrik Rugli. Cones and gauges in coniplex spaces: Spectral gaps and complex Perromi Frobenius theor y . Preprint 2007, Ami of Math., to appear.
6
ACHIEVEMENTS OF THE PROJECT:
- Research visit(s) to other institution(s).
- Outreach activities related to the projects main topic.
- Any other contribution, not addressed elsewhere, that you consider irnportant.
The maximum length for this section is 1 page. (Anal or Verdana, font
size lo).
1 was one of the organizer of the conferences:
1. Dynainical System II in Denton, USA, May 17-23, 2009
2. Dynarnical Systems Days, Antofagasta, Chile December 3-14 2007.
3. 1 was also a coorclinator of the session of Dynamical Systems duririg the conference COMCA 2009 in Antofagasta, Chile, August 2009.
1 have visted the following Institution.
1. Polish Academy of Sciences, Warsaw, Jun to Aug 2007 1 was invited by Prof.
Janina Kotus and by Prof. F. Przytycki. We conducted research on dynamics
of meromorphic functions.
2. Froni Mar to May 2007 1 visited University of North Texas, USA. With Prof.
M. Urbanski we did a research on random dynamics.
3. University of North Texas (twice, 2008). 1 was working there with M. Urbanski
on random transformations.
4. PUC (2008 twice). The first time 1 was mostly working with J. Riviera on
Collet-Eckmann transformations and with G. Iommi on thermodynamic formalism for sorne non-continuous maps. The second time 1 visted PUC wheii
M. Urbanski was there (he has spent 2 weeks in my University and two weeks
in PUC). We were working together with J. Rivera on Collet-Eckmanri transforrnations (both deterministic and random).
5. North Texas University, Denton, June 10 May 30, 2009. With M. Urbanski
we did research on Random Dynamical Systems, Meromorphic Dynamics anci
Random Transcendental Dynamics.
With my PhD student I. Inoquio we were working on her paper Therrnodynamic
Formalism for Transcendental maps, symbolic dynamic outlook. This is another
point of view at dynamics of meromorphic functions.
1
RESUMEN:
The goal of this project was to study the existence of invariant measure for meromorphic function. The project has two important direction. Thc first one was
deterministic, that is Wc consider a fixed ineromnorphic non-ratioiial function and we
try to see if one can fiud an iiivariant measure ahsolutely continuous with time respect
to Lebesgue measure, or, if the Julia set is not an entire sphere, with the respect
t.o a Hausdorff ineasure iii the Hausdorff dimension, which is normally equivalent
to time conformal measure. Another direction was take a different transformation
each time we iterate. \Ve a.ssume that the transformation we take depends on sorne
probabilistic rule. One can think that we are adding a white noise to a deterrninistic
transformation.
In the deterministic case the most important result was obtained for an exponential
map f(z)=a exp(z), where a is a non-zero cornplex naumber. With N. Dobbs we
proved that, if aii exponential mimap satisfies so-called Misiurewicz condition, then
there does not exist an invariant probability. This is a bit surprise since in general simple transformation which satisfies this condition llave this kind of measure.
Another result was that, for transformation of the form f(z)=R(exp(z)) where R
is a rational map, and f does not llave a finite asymptotic values, we proved tliat
hyperbolic maps llave a conformal measure and an absolutely continuous invariant
measures absolutely continuous with the respect to these measures.
In thc randomn case, with V. Mayer and M. Urbanski we first definicd mneasurahie
expanding random systems, then we develop the thermodynamical formalismn anci
establish, in particular, exponential dccay of correlations and analyticity of the expected pressure although the spectral gap property does not hoid. This theory is
then used to investigate fractal properties of conformal random systems. We prove
a Bowen's formula and develop the multifractal formalism of the Gibbs states. Depending oii the behavior of the Birkhoff sums of time pressure function we get a
natural classifications of the systems into two classes: quasi-deterrninistic systems
which share many properties of deterministic ones and essential random systems
which are rather generic and never bi-Lipschitz equivalent to deterministic systems.
Wc show iii the esseritial case that the Hausdorff measure vanishes which refutes a
conjecture of Bogenschutz aud Ochs. Wc finally give applications of our results to
various specific conforrnal random systems and positively answer a question of Bruck
and Burger conceruing time Hausdorff diinensiomm of random Julia sets. Moreover, with
G. Iomrni in sorne special case of random expansion of real number we ga ye a direct
formula for the Husdorff dimnension of level sets, t.lmat is the set of points witim given
frequency of random digits.
1
COOPERACIÓN INTERNACIONAL
7080227
N° Proyecto:
Nombre Colaborador (a) Extranjero (a): NEIL DOBBS
POLISH ACADEMY OF SCIENCES
Afiliación Institucional Actual:
Fechas de estadía
Desde :2510912008
Hasta :10110/2008
Describa las actividades realizadas y resultados obtenidos. Destaque su contribución al logro de los objetivos del proyecto. Si es
pertinente, indique las publicaciones conjuntas generadas, haciendo referencia a lo informado en la etapa Productos. Agregue en la
-
i
With Neil Dobbs we were studying the existence of invariant measures absolutely continuous with respect to the Lebesgue measure.
Thc existence of this measure is know for Misiurewicz type transformations and not existence for exponential maps with the orbit of
zero of the first return map to the Ieft haif plane escaping very fast towards infinity.
Let v_n IT'n(0)I where T is the return map to the Ieft half of the plane. Wc proved that, if v_n > O(log n), then absolutely
contifluous invariant measures cannot exist, and if v_n < O(log log n) and vn > d>l for all large n, then a absolutely continuous
invariant measure exist, however is not finite. In the preve of the theorem we used techniques similar to studying the Brownian
motion. It is however still remains open ifone can find a condition WhiCh can guarantee that one of the both cases exist.
Neil Dobbs also gay e two talks related to his work on non-differentiability of the prcssure function. The audience of the lectures was
our faculty and PhD and master students.
PRODUCTOS
ARTÍCULOS
Para trabajos en Prensa' Aceptados/Enviados adjunte copia de carta de aceptación o de recepción.
N
Coiculescu, 1.; Skorulski, B.
Autor (a)(es/as) :
Nombre Completo de la Revista: Monatshefte fur Mathematik
Thermodynamic formalism of transcendental entire maps of finite type
Título (Idioma original):
LS!
Indexación :
0026-9255
!SSN:
2007
Año:
152
Vol.:
2
N°:
105-123
Páginas:
Estado de la publicación a la fecha : Publicada
Otras Fuentes de financiamiento, si las hay
[iish Science Foundation (KBN)
Envía documento en papel :
Archivo Asociado al artículo : http:/,I evalcyt.coni cyt.c 1/ informe
si
CoiSko07a.pdf
1920321/II 060538!2008/50161
2
Coiculescu 1.; Skorulski B.
Rocky Mountain Journal of Mathematics
Perturbations in the Speiser class
ISI
Indexación :
0035-7596
ISSN :
2007
Año:
37
Vol. :
3
N°:
763-800
Páginas :
:
Publicada
fecha
a
la
publicación
Estado de la
Otras Fuentes de financiamiento, si las hay
Autor (a)(eslas) :
Nombre Completo de la Revista:
Título (Idioma original) :
Polish Scicnce Foundation (KBN)
Envía documento en papel:
Archivo Asociado al artículo:
si
3
N°:
Dobbs N; Skorulski B.,
Autor (a)(es/as) :
Nombre Completo de la Revista : Fundamenta Mathematicae
Non-existence of absolutely continuous invariant probabilities for exponential maps
Título (Idioma original):
ISI
Indexación :
0016-2736
ISSN :
2008
Año:
198
Vol.:
3
283-287
Páginas:
Estado de la publicación a la fecha : Publicada
Otras Fuentes de financiamiento, si las hay
Research Network on Low Dimensional Dynamics, PBCT ACT 17, CONICYT, Chile, EU Research Training Network
"Conforma] Structures and Dynamics"
Envía documento en papel:
Archivo Asociado al artículo
Autor (a)(es/as)
Nombre Completo de la Revista
Título (Idioma original):
Indexación
ISSN:
Año:
Vol.:
SI
4
Mayer, V.; Skorulski, B.; Urbanski, M.
Annais of Mathematics
Distance expanding random mappings, thermodynamic formalism, Gibbs measures and
fractal geometry
'SI
Páginas:
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Research Network on Low Dimensional Dynamics, PBCT ACT 17, CONICYT, Chile
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1920321/11060538/2008/5019/
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N°:
lommi, G.; Skorulski, B.
Autor (a)(es/as)
Proceedings of the AMS
Nombre Completo de la Revista
Hausdorff Dimension of Cantor Series
Título (Idioma original):
'SI
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Research Network on Low Dimensional Dynamics
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IS.pdf
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OTRAS PUBLICACIONES
Sin información ingresada.
CONGRESOS
Autor (a)(es/as)
Skorulski, B.
Título (Idioma original):
infinity
Nombre del Congreso:
Dynamics of sorne meromorphic functions with an asymptotic value eventually mapped onto
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POLONIA
Warsaw
31/07/2007
03/08/2007
First Joint International Meeting between thc AMS and the PTM
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no
pl.pdf
http:/!evalcyt.coflicyt.cI/infolTfleacademicO/ifldexPhp/iflVestlgadot!f4congresos/descarga/21920321111060538/2008/7393,'
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2
Bartlomiej Skorulski
Dynamics of sorne meromorphic functions
Conforma] Structures and Dynamics. The current state-of-art and perspectives
REINO UNIDO DE GB E IRLANDA DEL NORTE
Coventry
11/06/2007
15/06/2007
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N" :
Autor (a)(es/as) :
3
Bartlomiej Skorulski
Título (Idioma original) :
Nombre del Congreso:
Exponential Misurewicz does not give acip
COMCA 2008
País:
CHILE
Iquique
30/07/2008
01/08/2008
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p3.pdf
http://evalcyt.conicyt.cl/informe academico/indcx.php/investigadOr/f4 congresos/descarga/21920321/11060538/2008/7441/
Autor (a)(es/as) :
4
Bartlorniej Skorulski
Título (Idioma original) :
Nombre del Congreso :
Conjuntos aleatorios
COMCA 2009
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CHILE
Antofagasta
05/0812009
07108/2009
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p4.pdf
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5
Autor (a)(es/as):
Bartlornicj Skorulski
Título (Idioma original):
fractal geometry
Nombre del Congreso :
Disance expanding random mappings, thermodynamic formalism, Gibbs measures and
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CHILE
Antofagasta
07/09/2009
11/09/2009
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Dynamical Systems at Valparaíso, a satellite conference of the III CLAM
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p5.pdf
htlp://evalcyt.conicyt.cl/inlbnnc acadcmico/index.php/investigador/f4_congresosldescargai2i92032 1/1 1060538/2008/7443/
N°:
Autor (a)(es/as) :
6
Bartlomiej Skorulski
Real-Analicity of the Hausdorffdimension of the Hausdorffdimension of the radial Julia set
Título (Idioma original) :
of sorne transcendental meromorphic functions
Vil SIMPOSIO CHILENO DE MATEMÁTICA SOCIEDAD DE MATEMATICA DE
Nombre del Congreso:
CHILE
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CHILE
Punta de Traica
07/11/2007
10/11/2007
si
p6.pdf
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TESIS/MEMORIAS
Resultados de analisis usados en dinamica comieja
Título de Tesis :
Nombre y Apellidos del(de la) Alumno(a) : Francisco Bravo
Nombre y Apellidos del(de la) Tutor(a) : Rivera, J,; Skorulski B.
Título Grado:
Institución:
Pregrado
UCN
País:
Ciudad :
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CHILE
Antofagasta
Terminada
01/03/2006
29/12/2006
si
2
N°:
El teorema de Perron-Frobenius y la generalizacion de Ruche
Título de Tesis :
Nombre y Apellidos del(de la) Alumno(a) : Oscar Santamaria
Nombre y Apellidos del(de la) Tutor(a): Rivera, J,; Skorulski B.
Título Grado :
Institución:
Magister
UCN
País:
Ciudad :
Estado de Tesis :
Fecha Inicio :
Fecha Término:
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Archivo Asociado
CHILE
Antofagasta
Terminada
01/0312007
29/06/2007
si
3
Análisis Multifractal de Medidas Autosimilares
Título de Tesis :
Adriana Tapia
la)
Alumno(a)
:
Nombre y Apellidos del(de
Nombre y Apellidos del(de la) Tutor(a) : Bartlomiej Skorulski
Título Grado :
Institución:
Magister
UCN
País:
Ciudad :
Estado de Tesis :
Fecha Inicio :
Fecha Término
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Archivo Asociado
CHILE
Antofagasta
En Ejecución
01/03/2008
N°:
Título de Tesis :
4
Teoría de la Dimensión, Formalismo Termodinámico y Formula de Bowen
si
Nombre y Apellidos del(de la) Alumno(a) : Mery Choque Valdez
Nombre y Apellidos del(de la) Tutor(a): Bartlomiej Skorulski
Titulo Grado:
Institución:
Magister
UCN
País:
Ciudad:
Estado de Tesis :
Fecha Inicio :
Fecha Término :
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Archivo Asociado
CHILE
Antofagasta
Terminada
03/11/2008
03/07/2009
si
5
N°:
Teoria de Nevanlinna
Título de Tesis :
Nombre y Apellidos del(de la) Alumno(a) : Sebastian Sarmiento
Nombre y Apellidos del(de la) Tutor(a) :
Bartlomiej Skorulski
Título Grado :
Institución:
Magister
UCN
País:
CHILE
Antofagasta
En Ejecución
01/12/2008
Ciudad :
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Fecha Término
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si
6
N°:
DIFFEOMORFISMOS PARCIALMENTE HIPERBOLICOS
Título de Tesis :
Nombre y Apellidos del(de la) Alumno(a) : Fabian Contreras Barraza
Nombre y Apellidos del(de la) Tutor(a): Carlos Vasquez
Título Grado :
Institución :
Magister
UCN
País:
CHILE
Antofagasta
Terminada
03/1112006
01108/2008
si
Ciudad :
Estado de Tesis:
Fecha Inicio :
Fecha Término :
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7
TEOREMA DE DE RHAM
Título de Tesis :
Nombre y Apellidos del(de la) Alumno(a) : Daniela Ceron Urzua
Nombre y Apellidos del(de la) Tutor(a): Richard Urzua
Título Grado:
Magister
Institución :
UCN
País:
Ciudad :
Estado de Tesis:
Fecha Inicio :
Fecha Término
Envía documento en papel:
Archivo Asociado:
CHILE
Antofagasta
En Ejecución
03/11/2006
si
8
La Ecuación de Tricomi, un Cálculo directo
Título de Tesis :
Nombre y Apellidos del(de la) Alumno(a) : Sebastian Sarmiento
Nombre y Apellidos del(de la) Tutor(a) : Bartlomiej Skorulski
Título Grado:
Institución:
Pregrado
UCN
País:
CHILE
Antofagasta
Terminada
02/03/2007
03/03/2008
Ciudad :
Estado de Tesis:
Fecha Inicio :
Fecha Término :
Envía documento en papel :
Archivo Asociado
si
9
Rectificación de ecuaciones diferenciales: Diferencibilidad respecto de las
condiciones iniciales.
Nombre y Apellidos del(de la) Alumno(a) : Juan Roberto Carmona Herrera
Título de Tesis :
Nombre y Apellidos del(de la) Tutor(a) : Bernardo San Martin
Título Grado :
Institución:
Pregrado
UCN
País:
CHILE
Antofagasta
En Ejecución
03/08/2009
Ciudad :
Estado de Tesis :
Fecha Inicio :
Fecha Término:
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si
ANEXOS
A continuación se detallan los anexos fisicos/papel que no se incluyen en el informe en formato PDF.
1. Confirmations of submissions of articles
2. Information about students